We solve the derandomized direct product testing question in the low acceptance regime, by constructing new high dimensional expanders that have no small connected covers. We show that our complexes have swap cocycle expansion, which allows us to deduce the agreement theorem by relying on previous work. Derandomized direct product testing, also known as agreement testing, is the following problem. Let X be a family of k-element subsets of [n] and let $\{f_s:s\to\Sigma\}_{s\in X}$ be an ensemble of local functions, each defined over a subset $s\subset [n]$. Suppose that we run the following so-called agreement test: choose a random pair of sets $s_1,s_2\in X$ that intersect on $\sqrt k$ elements, and accept if $f_{s_1},f_{s_2}$ agree on the elements in $s_1\cap s_2$. We denote the success probability of this test by $Agr(\{f_s\})$. Given that $Agr(\{f_s\})=\epsilon>0$, is there a global function $G:[n]\to\Sigma$ such that $f_s = G|_s$ for a non-negligible fraction of $s\in X$ ? We construct a family X of k-subsets of $[n]$ such that $|X| = O(n)$ and such that it satisfies the low acceptance agreement theorem. Namely, $Agr (\{f_s\}) > \epsilon \; \; \longrightarrow$ there is a function $G:[n]\to\Sigma$ such that $\Pr_s[f_s\overset{0.99}{\approx} G|_s]\geq poly(\epsilon)$. A key idea is to replace the well-studied LSV complexes by symplectic high dimensional expanders (HDXs). The family X is just the k-faces of the new symplectic HDXs. The later serve our needs better since their fundamental group satisfies the congruence subgroup property, which implies that they lack small covers. We also give a polynomial-time algorithm to construct this family of symplectic HDXs.

翻译：我们通过构造没有小连通覆盖的新型高维扩张子，解决了低接受度机制下的去随机化直接积检验问题。我们证明了所构造的复合体具有交换余循环扩张性质，从而可基于先前工作推导出一致性定理。所谓去随机化直接积检验（亦称一致性检验）研究如下问题：设X是[n]的k元子集族，$\{f_s:s\to\Sigma\}_{s\in X}$是定义在子集$s\subset [n]$上的局部函数系综。考虑如下一致性检验：随机选取一对交集大小为$\sqrt k$的集合$s_1,s_2\in X$，若$f_{s_1},f_{s_2}$在交集$s_1\cap s_2$上一致则判定通过。记此检验成功概率为$Agr(\{f_s\})$。给定$Agr(\{f_s\})=\epsilon>0$，是否存在全局函数$G:[n]\to\Sigma$使得对不可忽略比例的$s\in X$有$f_s = G|_s$？我们构造了一个k元子集族X满足$|X| = O(n)$且具有低接受度一致性定理性质，即$Agr (\{f_s\}) > \epsilon$蕴含存在函数$G:[n]\to\Sigma$使得$\Pr_s[f_s\overset{0.99}{\approx} G|_s]\geq poly(\epsilon)$。关键思想是用辛高维扩张子（HDXs）替代广泛研究的LSV复形。该族X正是新型辛HDXs的k维面。后者因其基本群满足同余子群性质（这意味着不存在小覆盖）而更契合我们的需求。我们还给出了构造该辛HDXs族的多项式时间算法。